From Einstein to Schücking: Mach’s Principle and Relativistic Cosmology David E. Rowe Mainz University On Solving Einstein’s Gravitational Field Equations Einstein’s Gravitational Field Equations from November 1915: Gµν = κ Tµν 1 wo G = R − g R µν µν 2 µν κ = − 8π G (G = Newton Constant) Einstein vs. Newton Before Black Holes: Hilbert’s Version of the Schwarzschild Metric Relativistic Cosmology: Background Events Conversations in Leiden, 1916-1917 Group Photo in Leiden, 1921 • A. S. Eddington, H. A. Lorentz (front) • Einstein, Ehrenfest, de Sitter (back) Albert Einstein and Willem de Sitter • Fall 1916: first meeting with de Sitter • Leiden’s astronomer a close associate of Lorentz und Ehrenfest • By 1916 all three were deeply engaged in Einstein’s new generalized theory of relativity • But de Sitter had a head start having studied the earlier version of Einstein and Grossmann from 1913! • In 1916 de Sitter corresponded with Einstein about gravitational radiation! Cosmological Combatants: De Sitter vs. Einstein De Sitter’s Early Critique and Einstein’s Reaction Einstein to de Sitter, 2 February 1917 “Presently I am writing a paper on the boundary conditions in gravitation theory. I have completely abandoned my idea on the degeneration of the g µ ν ’s, which you rightly disputed. I am curious to see what you will say about the rather outlandish conception I have now set my sights on.” Einstein to Paul Ehrenfest, 4 February 1917 “I have again perpetrated something in gravitational theory that puts me in a bit of danger of being committed to a madhouse. I hope you don’t have any in Leiden so that I can visit you again safely.” Einstein’s Cosmological Field Equations from 1917 The cosmological equations are: G − λg = T µν µν µν where the tiny constant λ corresponds to a small repelling force. On Avoiding Boundary Conditions at Spatial Infinity • Einstein and Jakob Grommer had tried to find global solutions of the original field equations. • Solutions were to be centrally symmetric, static, and with appropriately chosen boundary values at spatial infinity. • DeSitter criticized this approach • Einstein went on to conclude that space must be finite, but unbounded. • By adding the cosmological constant Einstein came up with his cylindrical universe. The „Einstein Universe“, 1917 • Einstein modified his field equations for gravitation to produce a Machian universe • This „model“ represented a static universe compatible with general relativity • Felix Klein dubbed it Einstein‘s „Cylindrical World“ De Sitter Announces an Alternative Cosmology De Sitter to Einstein, 20 March, 1917 “I have found that the equations Gµ ν − λ gµν = 0 , thus your equations without matter, can be satisfied by [the de Sitter metric] . ..” “I do not know if it can be said that ‘inertia is explained’ in this way. I do not concern myself with explanations. If a single test particle existed in the world, where there was no sun and stars, it would have inertia.” Modern Picture of de Sitter‘s World with Static Coodinates De Sitter Spacetime admits Spatial Sections of Positive, Zero, and Negative Curvature How a de Sitter Model can reveal an „Expanding Universe“ Taking a Glance Backward: Historical Background On Machian Ideas in Einstein’s Theory of Gravitation Two Early Theoretical Contexts • Original context: Mach’s critique of Newtonian mechanics • New context: field physics 1) Einstein‘s appeal to the principle of relativity in electrodynamics; 2) coupling of mass and energy in SR; 3) coupling of inertia and gravity in GR Mach’s Critique of Newtonian Mechanics • Ernst Mach famously rejected Newton’s notions of absolute space and absolute time • Inertial properties of matter were due to some kind of interaction with distant masses (the fixed stars) Newton’s Principia (1687) Newton’s Rotating Water Bucket Mach’s Influence on Einstein • Problem of Rotation played a key role in Einstein’s efforts to extend the principle of relativity to non- inertial frames • In SR he showed how small inertial effects could be produced by rotating masses “Gibt es eine Gravitationswirkung, die der elektrodynamischen Induktionswirkung analog ist?” Einstein‘s first attempt to implement a Machian Program, 1912 Einstein in pursuit of a Machian approach to Gravitation (1912) • 1913: Einstein writes to Mach about the relativity of inertia • 1918: Hans Thirring carries out similar ideas in GR Einstein to Mach, June 1913 "it... turns out that inertia originates in a kind of interaction between bodies, quite in the sense of your considerations on Newton's pail experiment... If one rotates [a massive shell of matter] relative to the fixed stars about an axis going through its center, a Coriolis force arises in the interior of the shell; that is, the plane of a Foucault pendulum is dragged around (with a practically unmeasurably small angular velocity)." Machian Motifs in Relativistic Cosmology Einstein’s “Cosmological Considerations” (1917) “In a consistent theory of relativity there can be no inertia relative to ‘space,’ but only an inertia of masses relative to one another. If, therefore, I have a mass at a sufficient distance from all other masses in the universe, its inertia must fall to zero.” Willem de Sitter as Skeptic • De Sitter took a skeptical view of Mach’s principle, which he saw as a metaphysical idea rather than a testable scientific hypothesis • He also saw himself as differing in this respect from Einstein, to whom he wrote: “Our differences in belief amount to this: whereas you have a definite belief, I am a skeptic.” (18 April, 1917) Einstein on Mach’s Principle, 1918 Einstein on Mach’s Principle, 1918 • Einstein explicitly articulated what he understood by Mach’s Principle in a short note, published in Annalen der Physik in May 1918 • Philosophers of science know this paper as Einstein’s reply to Kretschmann, who claimed general covariance was a purely formal principle without physical significance (a point Einstein conceded) • Einstein replied that GR was a theory about space-time coincidences • But what about Mach’s Principle? Einstein’s Machian Views, ca. 1918 • In his 1918 note, Einstein named three principles as foundations for GR: a) Principle of Relativity, b) Principle of Equivalence, c) Mach’s Principle: the G-field is completely determined by the energy tensor of matter • In a footnote, he stated that until now he had not clearly separated a) and c), the latter being a generalization of Mach’s ideas about the origin of inertia Status of the Three Principles • a) just asserts that laws of physics can be expressed in terms of generally covariant equations, a useful heuristic guideline • b) was central to Einstein’s approach from 1907 • c) “is a different story. The necessity to uphold it is by no means shared by all colleagues, but I myself feel it is absolutely necessary.” • The remainder of the paper explains how c) led him to modify the field equations by introducing the cosmological constant, etc. Mach’s Principle: the Context of Articulation • Einstein wrote this reply to Kretschmann’s critique on 6 March, 1918 • A closer reading indicates that he mainly used it to present arguments bolstering his case for Mach’s principle and its relevance for cosmology • Precisely one day later, on 7 March, he submitted a short notice to the Prussian Academy entitled “Kritisches zu einer von Hrn. De Sitter gegebenen Lösung der Gravitationsgleichungen” De Sitter’s Spacetime as a Threat to Mach’s Principle Einstein (1918): “If the De Sitter solution were valid everywhere, it would show that the introduction of the ‘λ-term’ does not fulfill the purpose I intended. Because, in my opinion, the general theory of relativity is a satisfying system only if it shows that the physical qualities of space are completely determined by matter alone. Therefore, no g µ ν -field can exist (that is, no spacetime continuum is possible) without matter that generates it.” Einstein’s Critique • Einstein noted that the metric tensor vanishes on the horizon, and then concluded: • “The De Sitter system does not look at all like a world free of matter, but rather like a world whose matter is concentrated entirely on the [horizon] surface. This could possibly be demonstrated by means of a limiting process . ..” • Not long afterward Hermann Weyl would provide the details for this argument. Weyl’s Raum-Zeit-Materie, First Edition, 1918 Hubert Goenner on Weyl’s Shifting Views on de Sitter Spacetime Hubert Goenner, “Weyl’s Contributions to Cosmology,” in Erhard Scholz, ed., Hermann Weyl’s Raum – Zeit – Materie and a General Introduction to his Scientific Work, DMV Seminar Band 30, Birkhäuser, 2001 How Felix Klein used Projective Geometry to Clarify the Issue of Singularities Klein to Einstein, 14 June 1918 Klein reacted to Einstein’s critique of de Sitter’s model: “I came to the conclusion that the singularity you noticed can simply be transformed away.” Klein considered pencils of hyperplanes intersecting the quadric at temporal infinity. This yields a 6-parameter transitive group mapping the spacetime to itself. But observers in different worlds experience strange difficulties when they try to communicate with one another. How Can 2 Observers Synchronize Watches in de Sitter Spacetime? Klein to Einstein, 14 June 1918 “It is amusing to picture how two observers living on the quasi-sphere with differing de Sitter clocks would squabble with each other. Each of them would assign finite ordinates to some of the events that for the other would be lying at infinity or that would even show imaginary time values.” Einstein to Klein, 20 June 1918 “You are entirely right. De Sitter’s world is, in and of itself, free of singularities and its spacetime points are all equivalent. My critical remark about de Sitter’s solution needs correction: a singularity-free solution for the gravitational field equations without matter does in fact exist.